Testing central assumptions of trait-based ecology
Appendix
2023-01-19
- 1 Background
- 2 Compare Models
- 2.1 Train Models
- 2.2 Model Performance
- 2.3 Model Results: Tree Age + Plant
Traits + Environmental Conditions
- 2.3.1 Build Model
- 2.3.2 Relative Importance
- 2.3.3 Partial Dependence
- 2.3.4 Performance
- 2.3.5 Interactions | Table
- 2.3.6 Interaction | Group | Test
- 2.3.7 Interaction | Group | Violin
- 2.3.8 Interaction | Group | Boxplot
- 2.3.9 Interaction | Group | Density
- 2.3.10 Interaction | Group | Top
- 2.3.11 Interactions | Plots
- 2.3.12 Group Relative Importance
- 3 Compare to GLM
1 Background
We hope to explore the relative influence of physical traits, environmental conditions and species identity on the growth rate of trees. A gradient boosted model seems like a good candidate for this work since they:
- Can effectively model interactions, without the modeler specifying them
- Do not drop rows with NAs - these rows are put into their own node (i.e., na.action = na.pass)
- Can effectively model non-linear relationships - these are quite likely in the relationship between growth rate and the predictors. The GBM can help us understand them, but it can also model them.
1.1 Extracting Principle Components for Environmental Traits
We, first, converted the environmental variables to principle
components as they were highly correlated. We visualized the PCA and
used the eginvectors to help figure which environmental condition best
explained that PC. There were 5 -
Soil Fertility, Light, Temperature, pH, Soil.Humidity.Depth, and Slope.
1.1.1 PC1-PC2
1.1.2 P3-PC4
1.1.3 PC5-PC6
1.1.4 Correlation on Plant Traits
We want to ensure that the plant traits are not correlated. Past work suggests that they are not easily represented using a PCA. So, we will not use the this feature reduction method.
1.2 About Gradient Boosted Models
A gradient boosted machine/model is a machine learning model that uses decision trees to fit the data.
A decision tree first starts with all of the observations, then, from the variables provided, it tries to figure out which variable split would result in the “purest” groupings of the data. So, in this case, it would try to place rows with higher growth rates in one node, and those with lower growth rates in another node.
GBMs are an ensemble of decision trees, nut they are fit
sequentially. We call GBMs an ensemble of weak learners as each
subsequent tree is an attempt to correct the errors of the previous
tree. Thus, while one tree, by itself, can not describe the
relationships, with the use of all the trees, we can. Below is a figure
by Bradly Bohemke that attempts to illustrate how each subsequent tree
improves the fit on the data. 
2 Compare Models
We compared the fit of three used a gradient boosted models to determine how environmental gradients and physical traits influence RGR:
- Model 1 (Plant Traits + Environmental Conditions + Tree Age): this model assumes that in additional to tree age, growth rates are best described by the traits of the tree and environmental conditions.
- Model 2 (Species Identity + Environmental Conditions + Tree Age): it is possible that species identity is a stronger predictor of growth rates than the traits of the tree. As such, in this model, we ignore plant traits and use species identity.
- Model 3 (Species Identity + Plant Traits + Environmental Conditions + Tree Age): here, we hypothesize that while plant traits and environmental conditions are influential, we may have greater predictive power if we also consider the species of the tree.
Though we present outputs for all three models below, we show that
the best model is Model 1, using caret::resamples. This
function allows us to iteratively build models using the training data
and measure performance each round. In the end, we have re-sampled
measures of each performance metric - r squared, RSME and MSE. Before
comparing model performance, however, we first train the models - this
is done to help in determining a range of parameters that would fit the
data best for caret::resamples.
2.1 Train Models
Below, we show the best parameters for the models, given the data. ### Model 1: Tree Age + Plant Traits + Environmental Conditions{.tabset}
## $model_id
## [1] "final_grid_model_148"
##
## $training_frame
## [1] "train.hex"
##
## $validation_frame
## [1] "valid.hex"
##
## $score_tree_interval
## [1] 10
##
## $ntrees
## [1] 10000
##
## $max_depth
## [1] 3
##
## $min_rows
## [1] 4
##
## $nbins
## [1] 32
##
## $nbins_cats
## [1] 256
##
## $stopping_rounds
## [1] 5
##
## $stopping_metric
## [1] "deviance"
##
## $stopping_tolerance
## [1] 1e-04
##
## $max_runtime_secs
## [1] 3574.007
##
## $seed
## [1] 1234
##
## $learn_rate
## [1] 0.05
##
## $distribution
## [1] "gaussian"
##
## $sample_rate
## [1] 0.28
##
## $col_sample_rate
## [1] 0.55
##
## $col_sample_rate_per_tree
## [1] 0.27
##
## $min_split_improvement
## [1] 0
##
## $histogram_type
## [1] "RoundRobin"
##
## $categorical_encoding
## [1] "Enum"
##
## $calibration_method
## [1] "PlattScaling"
##
## $x
## [1] "Soil.Fertility" "Light" "Temperature" "pH"
## [5] "Slope" "Estem" "Branching.Distance" "Stem.Wood.Density"
## [9] "Leaf.Area" "LMA" "LCC" "LNC"
## [13] "LPC" "d15N" "t.b2" "Ks"
## [17] "Ktwig" "Huber.Value" "X.Lum" "VD"
## [21] "X.Sapwood" "d13C" "Tree.Age" "julian.date.2011"
##
## $y
## [1] "BAI_GR"2.1.1 Model 2: Tree Age + Species Identity + Environmental Conditions
## $model_id
## [1] "final_grid_model_12"
##
## $training_frame
## [1] "train.hex"
##
## $validation_frame
## [1] "valid.hex"
##
## $score_tree_interval
## [1] 10
##
## $ntrees
## [1] 10000
##
## $max_depth
## [1] 11
##
## $min_rows
## [1] 4
##
## $nbins
## [1] 256
##
## $nbins_cats
## [1] 4096
##
## $stopping_rounds
## [1] 5
##
## $stopping_metric
## [1] "deviance"
##
## $stopping_tolerance
## [1] 1e-04
##
## $max_runtime_secs
## [1] 3594.991
##
## $seed
## [1] 1234
##
## $learn_rate
## [1] 0.05
##
## $distribution
## [1] "gaussian"
##
## $sample_rate
## [1] 0.82
##
## $col_sample_rate
## [1] 0.7
##
## $col_sample_rate_per_tree
## [1] 0.92
##
## $min_split_improvement
## [1] 0
##
## $histogram_type
## [1] "QuantilesGlobal"
##
## $categorical_encoding
## [1] "Enum"
##
## $calibration_method
## [1] "PlattScaling"
##
## $x
## [1] "Soil.Fertility" "Light" "Temperature" "pH" "Slope"
## [6] "Species" "Tree.Age" "julian.date.2011"
##
## $y
## [1] "BAI_GR"2.1.2 Model 3: Tree Age + Species Identity + Plant Trait + Environmental Conditions
## $model_id
## [1] "final_grid_model_77"
##
## $training_frame
## [1] "train.hex"
##
## $validation_frame
## [1] "valid.hex"
##
## $score_tree_interval
## [1] 10
##
## $ntrees
## [1] 10000
##
## $max_depth
## [1] 6
##
## $min_rows
## [1] 2
##
## $nbins
## [1] 32
##
## $nbins_cats
## [1] 4096
##
## $stopping_rounds
## [1] 5
##
## $stopping_metric
## [1] "deviance"
##
## $stopping_tolerance
## [1] 1e-04
##
## $max_runtime_secs
## [1] 3547.872
##
## $seed
## [1] 1234
##
## $learn_rate
## [1] 0.05
##
## $distribution
## [1] "gaussian"
##
## $sample_rate
## [1] 0.2
##
## $col_sample_rate
## [1] 0.7
##
## $col_sample_rate_per_tree
## [1] 0.85
##
## $min_split_improvement
## [1] 1e-08
##
## $histogram_type
## [1] "RoundRobin"
##
## $categorical_encoding
## [1] "Enum"
##
## $calibration_method
## [1] "PlattScaling"
##
## $x
## [1] "Soil.Fertility" "Light" "Temperature" "pH"
## [5] "Slope" "Estem" "Branching.Distance" "Stem.Wood.Density"
## [9] "Leaf.Area" "LMA" "LCC" "LNC"
## [13] "LPC" "d15N" "t.b2" "Ks"
## [17] "Ktwig" "Huber.Value" "X.Lum" "VD"
## [21] "X.Sapwood" "d13C" "Species" "Tree.Age"
## [25] "julian.date.2011"
##
## $y
## [1] "BAI_GR"2.2 Model Performance
Here, we present results from model comparisons and show that the best model is Model 1 - Plant Traits + Environmental Conditions.
2.2.1 Summary
##
## Call:
## summary.resamples(object = ModelPerformanceCompare)
##
## Models: Model 1 - Plant Traits + Environmental Conditions, Model 2 - Species Identity + Environmental Conditions, Model 3 - Species Identity + Environmental Conditions + Plant Traits
## Number of resamples: 25
##
## MAE
## Min. 1st Qu. Median
## Model 1 - Plant Traits + Environmental Conditions 0.6207895 0.6957431 0.7265563
## Model 2 - Species Identity + Environmental Conditions 0.7598406 0.8467461 0.8879756
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 0.7008084 0.7904407 0.8166411
## Mean 3rd Qu. Max. NA's
## Model 1 - Plant Traits + Environmental Conditions 0.7408249 0.7904460 0.8598637 0
## Model 2 - Species Identity + Environmental Conditions 0.8964071 0.9432935 1.0775755 0
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 0.8183526 0.8591769 0.9161170 0
##
## RMSE
## Min. 1st Qu. Median
## Model 1 - Plant Traits + Environmental Conditions 0.8024465 0.9154721 1.014256
## Model 2 - Species Identity + Environmental Conditions 1.0508989 1.1441543 1.214355
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 0.9273804 1.0402613 1.092019
## Mean 3rd Qu. Max. NA's
## Model 1 - Plant Traits + Environmental Conditions 1.012689 1.061483 1.250678 0
## Model 2 - Species Identity + Environmental Conditions 1.214074 1.258653 1.419486 0
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 1.096026 1.125715 1.333776 0
##
## Rsquared
## Min. 1st Qu. Median
## Model 1 - Plant Traits + Environmental Conditions 2.158172e-02 0.093931513 0.13196451
## Model 2 - Species Identity + Environmental Conditions 1.349711e-05 0.002559265 0.01320337
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 4.107481e-04 0.021497133 0.07528740
## Mean 3rd Qu. Max.
## Model 1 - Plant Traits + Environmental Conditions 0.13189257 0.17315454 0.2605188
## Model 2 - Species Identity + Environmental Conditions 0.02671596 0.02832347 0.1401977
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 0.07068428 0.10458316 0.1884573
## NA's
## Model 1 - Plant Traits + Environmental Conditions 0
## Model 2 - Species Identity + Environmental Conditions 0
## Model 3 - Species Identity + Environmental Conditions + Plant Traits 02.2.2 Boxplot
2.2.3 Dotplot
2.3 Model Results: Tree Age + Plant Traits + Environmental Conditions
2.3.1 Build Model
Now, we can build the model.
set.seed(123)
gbm_regressor_bai_residuals <-
gbm(BAI_GR ~ .,
data =
rgr_msh_na %>% filter(Group == "Train")%>% filter(!is.na(BAI_GR))%>%
select(any_of(c(EnvironmentalVariablesKeep,
PlantTraitsKeep, "Tree.Age", "BAI_GR", "julian.date.2011"))),
n.trees = 1000,
interaction.depth = 3, #max depth
shrinkage = 0.05, #learning rate
n.minobsinnode = 13, #col_sample_rate
bag.fraction = 0.28, # sample_rate,
verbose = FALSE,
n.cores = NULL,
cv.folds = 5)2.3.2 Relative Importance
First, we look at the importance of variables in the model.
2.3.3 Partial Dependence
Assessing how, when we hold everything else constant, what the relationships are between growth rate and the predictor.
Soil Fertility
Light
Temperature
pH
Slope
Modulus of Elasticity for Stem
Branching Distance
Stem Wood Density
Leaf Area
Leaf Mass Per Area
Leaf Carbon Concentration
Leaf Nitrogen Concentration
Leaf Phosphorus Concentration
Delta 15N
Thickness to Span Ratio
Conductivity Per Sapwood Area
Conductivity per Branch
Huber Value
Percent Lumen
Vessel Diameter
Percent Sapwood
Delta 13C
Tree Age
Julian Date in 2011
2.3.4 Performance
2.3.5 Interactions | Table
Let’s explore the interactions in these data.
2.3.6 Interaction | Group | Test
##
## Kruskal-Wallis rank sum test
##
## data: Value by Class
## Kruskal-Wallis chi-squared = 22.488, df = 2, p-value = 1.309e-05## Comparison Z
## 1 Environmental Conditions:Environmental Conditions - Plant Traits:Environmental Conditions -3.422308
## 2 Environmental Conditions:Environmental Conditions - Plant Traits:Plant Traits -4.679483
## 3 Plant Traits:Environmental Conditions - Plant Traits:Plant Traits -1.405564
## P.unadj P.adj
## 1 6.209183e-04 1.241837e-03
## 2 2.875992e-06 8.627977e-06
## 3 1.598537e-01 1.598537e-012.3.7 Interaction | Group | Violin
2.3.8 Interaction | Group | Boxplot
2.3.9 Interaction | Group | Density
2.3.10 Interaction | Group | Top
2.3.11 Interactions | Plots
Now, we plot interactions with values>0.10.
Leaf Phosphorus Concentration:Conductivity per Branch
Leaf Nitrogen Concentration:Conductivity Per Sapwood Area
Huber Value:Tree Age
Soil Fertility:Stem Wood Density
Delta 15N:Huber Value
Vessel Diameter:Percent Sapwood
Conductivity Per Sapwood Area:Julian Date in 2011
Slope:Vessel Diameter
Light:Delta 13C
Percent Lumen:Percent Sapwood
Soil Fertility:Huber Value
Soil Fertility:Percent Sapwood
Soil Fertility:Leaf Carbon Concentration
Branching Distance:Leaf Nitrogen Concentration
Leaf Nitrogen Concentration:Vessel Diameter
Leaf Area:Percent Sapwood
Stem Wood Density:Delta 15N
Leaf Area:Tree Age
Slope:Tree Age
Thickness to Span Ratio:Conductivity per Branch
pH:Conductivity per Branch
Delta 15N:Conductivity per Branch
Leaf Area:Thickness to Span Ratio
Leaf Carbon Concentration:Percent Sapwood
Stem Wood Density:Leaf Carbon Concentration
Modulus of Elasticity for Stem:Huber Value
Light:Branching Distance
Conductivity per Branch:Julian Date in 2011
Leaf Phosphorus Concentration:Thickness to Span Ratio
Stem Wood Density:Huber Value
Light:Julian Date in 2011
Branching Distance:Percent Lumen
Thickness to Span Ratio:Conductivity Per Sapwood Area
Modulus of Elasticity for Stem:Branching Distance
pH:Tree Age
Light:Tree Age
Leaf Nitrogen Concentration:Percent Lumen
2.3.12 Group Relative Importance
Finally, we compare the relative importance of the various groups - tree age, plant traits, and environmental conditions.
3 Compare to GLM
The model below is a GLM with the main effects and top 22 interactions from the best model. This will be used as a baseline to compare to other models Julie found in literature. We caution that this is an exploratory model, and is not a true translation of the GBM as we do not model non-linearity.
##
## Call:
## glm(formula = TestModelFormula, data = rgr_msh_na)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9145 -0.5267 -0.1461 0.3479 3.4778
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.958e+00 3.367e+00 1.770 0.07822 .
## Soil.Fertility 7.276e-01 1.400e+00 0.520 0.60372
## Light 1.675e+00 1.492e+00 1.123 0.26266
## Temperature 7.468e-02 9.302e-02 0.803 0.42293
## pH -1.725e-01 1.536e-01 -1.123 0.26277
## Slope -9.130e-03 1.553e-01 -0.059 0.95318
## Estem 1.961e-05 2.406e-05 0.815 0.41592
## Branching.Distance -2.833e-02 1.644e-02 -1.724 0.08620 .
## Stem.Wood.Density 5.656e-01 9.061e-01 0.624 0.53318
## Leaf.Area 6.877e-04 4.916e-03 0.140 0.88888
## LMA 1.925e-02 5.804e-03 3.316 0.00107 **
## LCC -3.369e-02 3.063e-02 -1.100 0.27266
## LNC -9.213e-03 2.649e-01 -0.035 0.97229
## LPC -1.404e-02 2.927e-02 -0.480 0.63206
## d15N -1.363e-01 1.540e-01 -0.885 0.37729
## t.b2 -1.047e+01 1.557e+01 -0.673 0.50171
## Ks -1.394e-02 1.288e-02 -1.082 0.28041
## Ktwig 1.488e-04 5.997e-04 0.248 0.80431
## Huber.Value 2.936e+00 5.575e+00 0.527 0.59898
## X.Lum -4.015e-01 7.229e+00 -0.056 0.95576
## VD 1.626e-04 7.483e-05 2.172 0.03094 *
## X.Sapwood -4.738e-01 1.180e+00 -0.401 0.68857
## d13C 1.420e-01 6.873e-02 2.066 0.04003 *
## Tree.Age -4.475e-03 1.027e-02 -0.436 0.66335
## julian.date.2011 3.263e-03 8.494e-03 0.384 0.70127
## LPC:Ktwig -8.664e-07 2.904e-05 -0.030 0.97623
## LNC:Ks 3.844e-03 2.099e-03 1.831 0.06843 .
## Huber.Value:Tree.Age -3.513e-03 1.821e-01 -0.019 0.98463
## Soil.Fertility:Stem.Wood.Density 8.891e-01 5.171e-01 1.719 0.08698 .
## d15N:Huber.Value -1.041e+00 7.993e-01 -1.302 0.19419
## VD:X.Sapwood 6.428e-05 5.786e-05 1.111 0.26787
## Ks:julian.date.2011 2.216e-06 6.299e-05 0.035 0.97196
## Slope:VD 1.377e-05 7.893e-06 1.744 0.08254 .
## Light:d13C 5.226e-02 4.916e-02 1.063 0.28901
## X.Lum:X.Sapwood -3.803e+00 8.882e+00 -0.428 0.66894
## Soil.Fertility:Huber.Value 1.196e+00 1.534e+00 0.779 0.43657
## Soil.Fertility:X.Sapwood -3.235e-03 3.966e-01 -0.008 0.99350
## Soil.Fertility:LCC -2.948e-02 2.934e-02 -1.005 0.31624
## Branching.Distance:LNC 1.390e-02 5.980e-03 2.325 0.02102 *
## LNC:VD -6.617e-05 2.835e-05 -2.334 0.02052 *
## Leaf.Area:X.Sapwood 1.531e-03 7.118e-03 0.215 0.82990
## Stem.Wood.Density:d15N 3.758e-01 2.250e-01 1.670 0.09630 .
## Leaf.Area:Tree.Age -1.297e-04 1.262e-04 -1.028 0.30512
## Slope:Tree.Age -3.017e-03 6.087e-03 -0.496 0.62063
## t.b2:Ktwig 1.021e-02 1.334e-02 0.765 0.44510
## pH:Ktwig -4.869e-05 1.024e-04 -0.475 0.63497
## d15N:Ktwig -2.259e-05 4.167e-05 -0.542 0.58827
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.8795727)
##
## Null deviance: 283.97 on 259 degrees of freedom
## Residual deviance: 187.35 on 213 degrees of freedom
## (2 observations deleted due to missingness)
## AIC: 748.64
##
## Number of Fisher Scoring iterations: 2## McFadden's R-squared for model is 0.34025566703557